Lets say we have a utility function $U: \mathbb{R}^{2} \to \mathbb{R}$ given by $U(x,y)$ and a binding budget constraint $p_{x} x + p_{y} y = m$, where $p_{x}, p_{y}$ are prices of goods $x,y$ and $m$ is income. We can maximize this function subject to our constraint by simply maximizing this the following function in the unconstrained sense: $$ \mathcal{L}(x,y, \lambda) = U(x,y) + \lambda(m -p_{x}x - p_{y}{x}) $$ where $\lambda$ is usually called the Lagrange Multiplier.
My question: The reason we're able to add the second term on the RHS of the expression above is because our budget constraint binds, i.e. $m - p_{x}x - p_{y}y = 0$, so we're essentially adding zero. But, why is the conventional expression inside the parantheses after $\lambda$ given by $m - p_{x}x - p_{y}y$ and not $p_{x}x + p_{y}y - m$? Shouldn't both expressions give us a zero term? My guess is that this has something to do with the sign of $\lambda$ (which we want to be positive because it can be interpreted as the marginal utility of $m$), but I'm not sure how exactly this works, mathematically. Would appreciate a clarification explained mathematically.
